# Limit problem from past exam.

limit problem from past exam. :(

## Homework Statement

Problem from an exam that I just did is really bugging me.

lim(x,y)->(0,0) (x*y2)/(x2+y4)

The attempt at a solution[/b]
At first I thought the limit didn't exist. I tried using paths, x=y1/2, but then I couldn't find a counter example for which a path approached at a different point. So then I tried using the squeeze theorem, and got that the limit is 0. :\

Was my approach to the problem correct? is the limit really 0?

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Dick
Homework Helper

You didn't say how you proved it. But yes, the limit is zero.

You didn't say how you proved it. But yes, the limit is zero.

Oh really? XD Well I assumed:

0<y2<x2+y4 (since x^2 and y^4 are always positive, and y2<y4

0<y2/x2+y4<1
Then multiplying x into the inequality:

0<x*y2/x2+y4<x

Then taking the limit:
lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y2/x2+y4<lim(x,y)->(0,0) x

Which gave me that:
0<lim(x,y)->(0,0) x*y2/x2+y4<0

which by the squeeze theorem, limit is zero. Was that correct...?

Dick
Homework Helper

Oh really? XD Well I assumed:

0<y2<x2+y4

0<y2/x2+y4<1
Then multiplying x into the inequality:

0<x*y2/x2+y4<x

Then taking the limit:
lim(x,y)->(0,0) 0<lim(x,y)->(0,0) x*y2/x2+y4<lim(x,y)->(0,0) x

Which gave me that:
0<lim(x,y)->(0,0) x*y2/x2+y4<0

which by the squeeze theorem, limit is zero. Was that correct...?
Well the starting point 0<y^2<x^2+y^4 is wrong. Suppose x=0 and y=(1/2). I would use polar coordinates.

Limit doesn't exist , try y=x^(1/2)

Aaah... I was starting it with polar coordinates at first and it seemed like the best way to go, but then it seemed to messy so I thought I was doing it wrong. I was running out of time and just left it like that. XD Thanks lots, at least maybe I have partial credit haha..

lim x->0 f(x,x^(1/2))=1/2
and lim x->0 f(x,0)=0

Dick
Homework Helper

Limit doesn't exist , try y=x^(1/2)
Ooops. You are so right. Sorry.

Aww darn... it was a 3 part question worth 5 points total on that part of the exam, so I should have 10/3 credit hehe... The rest of the test was pretty good.